A Reduced-Intensity Conditioning Regimen (Cyclophosphamide, Pentostatin, Anti-thymocyte Globulin) Followed by Haploidentical Hematopoietic Stem Cell Transplant for the Treatment of Patients With Refractory or Recurrent Severe Aplastic Anemia

Inclusion Criteria: * RECIPIENT: Documented informed consent of the participant. * RECIPIENT: Age: \>= 40 years but =\< 75 years of age at time of enrollment. * RECIPIENT: Failed at least one trial of immunosuppressive therapy (IST) by being refractory (persistence of severe cytopenia and fulfillment of SAA disease criteria at least 3 months after initial IST) or having relapsed (initial improvement of cytopenia after first-line IST but then a later return to fulfillment of SAA disease criteria when IST is decreased or ceased). IST could have included anti-thymocyte globulin (ATG) based regimens, calcineuPririn inhibitors and/or other higher dose therapy directed at the treatment of primary SAA. * RECIPIENT: Karnofsky performance score \>= 60%. * RECIPIENT: Confirmed diagnosis of severe aplastic anemia as: * Bone marrow cellularity \< 25% or marrow cellularity \< 50% but with \< 30% residual hematopoietic cells; * Two out of three of the following (in peripheral blood): * Neutrophils \< 0.5 x 10\^9/L; * Platelets \< 20 x 10\^9/L; * Reticulocyte count \< 20 x 10\^9/L (\< 60 x 10\^9/L using an automated analysis). * RECIPIENT: No suitable fully matched related sibling donor (6/6 match for human leukocyte antigen \[HLA\]-A and B at intermediate or high resolution and DRbetaA1 at high resolution using deoxyribonucleic acid \[DNA\]-based typing) available. * RECIPIENT: Available relative of the patient who is a haploidentical match, including biological parents, siblings or half siblings, children, uncles/aunts, first cousins, etc. Eligible haploidentical donors will have 2-4 mismatches if HLA-A, -B, -C, and -DRB1 typing is used; 2-5 mismatches if HLA-A, -B, -C, -DRB1, and -DQB1 typing is used; and 2-6 mismatches if HLA-A, -B, -C, -DRB1, -DQB1, and -DPB1 typing is used. A unidirectional mismatch in either the graft versus host or host versus graft direction is considered a mismatch. The donor and recipient must demonstrate that they are a full haplotype match by being identical at a minimum of one allele (at high resolution DNA-based typing) at the following genetic loci: HLA-A, -B, -C, and DRB1 if 8 allele typing is used; HLA-A, -B, -C, -DRB1, and -DQB1 if 10 allele typing is used; and HLA-A, -B, -C, -DRB1-, DQB1, and -DPB1 is 12 allele typing is used by the local center. * RECIPIENT: Patient and/or legal guardian must sign informed consent for the hematopoietic stem cell transplantation (HSCT). * RECIPIENT: The haplo donor and/or legal guardian must be able to sign informed consent documents. * RECIPIENT: The potential haplo donor must be willing and able to donate bone marrow. * RECIPIENT: The weight of the haplo donor must be \>= 20 kg. * RECIPIENT: Total bilirubin \< 3.0 x the upper limit of normal (ULN) for age (patients who have been diagnosed with Gilbert's Disease are allowed to exceed this limit) * RECIPIENT: Aspartate aminotransferase (AST) =\< 5.0 x ULN. * RECIPIENT: Alanine transaminase (ALT) =\< 5.0 x ULN. * RECIPIENT: * For patients \>= 13.0 years of age at the time of enrollment: Creatinine clearance of \>= 50 mL/min per 24 hour urine test or the Cockcroft-Gault formula. * For patients \< 13 years of age at enrollment: Glomerular filtration rate (GFR) estimated by the updated Schwartz formula \> 90 mL/min/1.73 m\^2. If the estimated GFR is \< 90 mL/min/1.73m\^2, then renal function must be measured by 24-hour creatinine clearance or nuclear GFR, and must be \> 50mL/min/1.73m\^2. * RECIPIENT: Women of childbearing potential (WOCBP): negative urine or serum pregnancy test If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. * RECIPIENT: Echocardiogram (ECHO) or multigated acquisition (MUGA): Left ventricular ejection fraction (LVEF) at rest \>= 40%. For patients aged \< 13 years, shortening fraction (SF) \>= 26% by echocardiogram or MUGA may be substituted for LVEF. * RECIPIENT: * For patients \> 13.0 years of age: Diffusing capacity of the lung for carbon monoxide (DLCO) (corrected/adjusted for hemoglobin) \> 40% and forced expiratory volume in 1 second (FEV1) \> 50% predicted (without administration of bronchodilator) and forced vital capacity (FVC) \> 50% predicted. * For patients \< 13.0 years of age unable to perform pulmonary function tests (PFTs) due to age or developmental ability: (1) no evidence of dyspnea at rest and (2) no need for supplemental oxygen and (3) oxygen (O2) saturation \> 92% on room air at sea level (with lower levels allowed at higher elevations per established center standard of care \[e.g., Utah, 4,200 feet above sea level, does not give supplemental oxygen unless below 90%\]). * RECIPIENT: Seronegative for human immunodeficiency virus (HIV) antigen and antibody (Ag/Ab) combo, hepatitis C virus (HCV), active hepatitis B virus (HBV) (surface antigen negative), and syphilis (rapid plasma reagin \[RPR\]) * If positive, hepatitis C ribonucleic acid (RNA) quantitation must be performed. * RECIPIENT: Antibodies to donor red blood cell antigens including ABO and rhesus (Rh) meets institutional titer requirements. * RECIPIENT: WOCBP or sexually active male: Agreement to use adequate contraception prior to study entry and 6 months post-CD4+ T-cell-depleted-HaploHCT. * DONOR: Documented protocol-specific City of Hope (COH) informed consent per local, state and federal guidelines * DONOR: Documented general institutional informed consent per local, state and federal guidelines. * DONOR: Age: younger than 60 years. * DONOR: Haplo donor selection is based on HLA typing and relationship to recipient. * DONOR: When more than one donor is available, the donor with the lowest number of HLA allele mismatches will be chosen unless there is HLA cross-match incompatibility or a medical reason to select otherwise. In cases where there is more than one donor with the least degree of mismatch, donors will be selected based on the most favorable combination of HLA compatibility in cross-match testing and ABO compatibility. Prioritization is given to the lowest number of mismatches in the host-versus-graft (HVG) direction to minimize the risk of graft failure. * DONOR: If there is more than one donor with the least amount of host-versus-graft (HVG) allele mismatches, the suggested prioritization in order of importance includes ABO compatibility, cytomegalovirus (CMV) status (use a sero-negative donor for a sero-negative recipient or use a sero-positive donor for a sero-positive recipient), younger age and lighter weight (this rule applies down to the age of 18, however, children may also be used as donors if appropriate), and sex of the donor (if all else is equal, males are preferred over nulliparous females over multiparous females). * DONOR: Infectious disease screening performed within 30 days prior to stem cell collection and per federal guidelines and is: * Seronegative for HIV Ag, HIV 1+2 Ab, human T-lymphotropic virus (HTLV) I/II Ab, hepatitis B virus surface antigen (HBsAg), hepatitis B virus core antibody (HBcAb) (immunoglobulin M \[IgM\] and immunoglobulin G \[IgG\]), HCV Ab; * Negative RPR for syphilis. * DONOR: WOCBP: Urine pregnancy testing performed within 7 days prior to stem cell mobilization * DONOR: Is approved and completed evaluation per institutional guidelines. Exclusion Criteria: * RECIPIENT: Inherited and acquired (non-aplastic anemia) bone marrow failure syndromes such as Fanconi anemia must be ruled out according to center standards. * RECIPIENT: Clonal cytogenetic abnormalities consistent with pre-myelodysplastic syndrome (pre-MDS) or MDS on marrow examination (e.g. Monosomy 7). * RECIPIENT: Any somatic mutations (including TERT, TERC, DKC1, NOP10) other than PIG-A (PNH) or BCOR mutation. * RECIPIENT: Diagnosis of myelodysplastic syndrome (MDS). * RECIPIENT: Presence of anti-donor HLA antibodies (positive anti-donor HLA antibody is defined as a positive cross-match test of any titer by complement-dependent cytotoxicity or flow cytometric testing or the presence of anti-donor HLA antibody to the high expression loci HLA-A, B, C, DRB1, or DPB1 with mean fluorescence intensity \[MFI\] \> 1000 by solid phase immunoassay). * RECIPIENT: Prior allogeneic stem cell transplant. * RECIPIENT: Prior solid organ transplant. * RECIPIENT: Known life-threatening reaction (i.e., anaphylaxis) to Thymoglobulin (registered trademark) that would prohibit use for the patient as this study requires use of the Thymoglobulin (registered trademark) preparation of ATG. * RECIPIENT: Uncontrolled bacterial, viral, or fungal infection at the time of enrollment. Uncontrolled is defined as currently taking medication and with progression or no clinical improvement on adequate medical treatment. * RECIPIENT: Seropositive for the human immunodeficiency virus (HIV). * RECIPIENT: Active hepatitis B or C determined by a detectable viral load of HBV or HCV. * RECIPIENT: Female patients who are pregnant (per institutional practice) or breast-feeding. * RECIPIENT: Prior malignancies except resected basal cell carcinoma or treated cervical carcinoma in situ. Cancer treated with curative intent \> 5 years previously will be allowed. Cancer treated with curative intent =\< 5 years previously will not be allowed unless approved by the protocol chairs and/or protocol officer. * RECIPIENT: Alemtuzumab or ATG within 2 weeks of enrollment. * RECIPIENT: Any other condition that would, in the investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures. * RECIPIENT: Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics). * DONOR: Has undergone any transplantation (i.e. organ, stem cell, bone marrow, blood). * DONOR: Receiving any investigational agents, or concurrent biological, chemotherapy, immunosuppression or radiation therapy. * DONOR: Active infection. * DONOR: Medical or physical reason which makes the donor unlikely to tolerate or cooperate with growth factor therapy and leukapheresis. * DONOR: Factors which place the donor at increased risk for complications from leukapheresis or granulocyte colony-stimulating factor (G-CSF) therapy. * DONOR: WOCBP: pregnant or =\< 6 months breastfeeding.