Clofarabine, Cytarabine, and Filgrastim Followed by Infusion of Non-HLA Matched Ex Vivo Expanded Cord Blood Progenitors in Treating Patients With Acute Myeloid Leukemia

Inclusion Criteria: * Cohort A: Diagnosis of AML by World Health Organization (WHO) criteria, either relapsed or refractory; acute promyelocytic leukemia \[Acute promyelocytic leukemia with t(15;17)(q22;q12) and variants\] will be eligible only after failure of a regimen containing arsenic trioxide; patients in this cohort must have had an initial remission duration of \< 1 year and cannot have received any prior salvage chemotherapy * Cohort B: Untreated AML patients, excluding those with favorable cytogenetic or molecular abnormalities per the European LeukemiaNet recommendations * Cohort C: Untreated AML patients, including those with favorable cytogenetic or molecular abnormalities per the European LeukemiaNet recommendations * The first three patients enrolled in cohorts A and B must be less than 60 years old; thereafter, patients aged \>= 18 and =\< 70 are eligible * The first three patients enrolled in cohorts A and B must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 -1; thereafter, ECOG performance status of 0-2 is required * Serum creatinine =\< 1.0 mg/dL; if serum creatinine \> 1.0 mg/dl, then the estimated glomerular filtration rate (GFR) must be \> 60 mL/min/1.73 m\^2 as calculated by the Modification of Diet in Renal Disease equation where predicted GFR (ml/min/1.73 m\^2) = 186 X (Serum Creatinine)\^-1.154 X (age in years)\^-0.203 X (0.742 if patient is female) X (1.212 if patient is black) * Serum total bilirubin =\< 1.5 x upper limit of normal (ULN) unless elevation is thought to be due to Gilbert's syndrome, hemolysis, or hepatic infiltration by the hematologic malignancy * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =\< 2.5 X ULN, unless elevation is thought to be due to hepatic infiltration by the hematologic malignancy * Alkaline phosphatase =\< 2.5 x ULN * Capable of understanding the investigational nature, potential risks and benefits of the study, and able to provide valid informed consent * Female patients of childbearing potential must have a negative serum pregnancy test within 2 weeks prior to enrollment * Male and female patients must be willing to use an effective contraceptive method during the study and for a minimum of 90 days after study treatment Exclusion Criteria: * Allogeneic transplant recipients * Current concomitant chemotherapy, radiation therapy, or immunotherapy other than as specified in the protocol with the exception of intrathecal chemotherapy administered on days that are not concurrent with clofarabine and cytarabine * Have any other severe concurrent disease, or have a history of serious organ dysfunction or disease involving the heart, kidney, liver (including symptomatic hepatitis, veno-occlusive disease), or other organ system dysfunction * Patients with a systemic fungal, bacterial, viral, or other infection not controlled (defined as exhibiting ongoing signs/symptoms related to the infection and without improvement, despite appropriate antibiotics or other treatment) * Pregnant or lactating patients * Any significant concurrent disease, illness, or psychiatric disorder that would compromise patient safety or compliance, interfere with consent, study participation, follow up, or interpretation of study results