Testing Atezolizumab With Selinexor in People ≥ 12 Years Old With Alveolar Soft Part Sarcoma, AXIOM Trial

Inclusion Criteria: * To be enrolled in the safety run-in, patients must have an advanced soft tissue sarcoma (not otherwise specified \[NOS\]). To be enrolled in the ASPS cohort, patients must have histologically or cytologically confirmed alveolar soft part sarcoma that is not curable by surgery * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as ≥ 20 mm (≥ 2 cm) by chest x-ray or as ≥ 10 mm (≥ 1 cm) with CT scan, MRI, or calipers by clinical exam. * Patients with unresectable, metastatic and measurable ASPS that is resistant/refractory to ICI treatment will be eligible for the ASPS cohort of this study if they show clinical and/or radiological evidence of disease progression after receiving prior ICI therapy (including history and increasing physical symptoms). On-study documentation will include a physician's rationale that supports evidence of clinical disease progression (i.e., increasing tumor pain) * Age \>= 12 years. Because no dosing or adverse event data are currently available on the use of atezolizumab in combination with selinexor in patients ˂ 18 years of age, children \< 12 years of age are excluded from this study * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 70%) * Absolute neutrophil count \>= 1,000/mcL * Platelets \>= 100,000/mcL * Hemoglobin \>= 9 g/dL * Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) (however, patients with known Gilbert disease who have serum bilirubin level =\< 3 x ULN may be enrolled) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) =\< 2.5 x institutional ULN or =\< 5 x ULN for patients with liver metastases * Serum creatinine =\< 1.5 x institutional ULN OR creatinine clearance \>= 30 mL/min/1.73 m\^2 by Cockcroft-Gault * Serum albumin \>= 2.5 g/dL * Baseline sodium (Na+) \>= 130 mEq/L * Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 28 days are eligible for this trial * Administration of selinexor or atezolizumab may have an adverse effect on pregnancy and poses a risk to the human fetus, including embryo-lethality. Female patients of child-bearing potential and male patients must agree to use highly effective contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, for 90 days after the last dose of selinexor, and for 150 days after the last dose of atezolizumab, whichever is longer. Breastfeeding is not allowed while on selinexor or for 90 days after the last dose of selinexor. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately * Ability to understand and the willingness to sign a written informed consent document * Willingness to provide biopsy samples for research purposes, except patients enrolled on the safety run-in. Patients that cannot be safely biopsied may be considered for the study upon discussion with principal investigator * Ability and willingness to swallow pills * Vaccines intended to prevent severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and coronavirus disease 2019 (COVID-19) are allowed Exclusion Criteria: * Malabsorption syndrome or other conditions that would interfere with intestinal absorption * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study. However, the following therapies are allowed: * Hormone-replacement therapy or oral contraceptives * Herbal therapy \> 1 week prior to cycle 1, day 1 (herbal therapy intended as anti-cancer therapy must be discontinued at least 1 week prior to cycle 1, day 1) * Palliative radiotherapy for bone metastases \> 2 weeks prior to cycle 1, day 1 * Treatment with any other agent administered for the treatment of the patient's cancer, within five half-lives or 3 weeks prior to cycle 1, day 1, whichever is shorter * History of malignancy other than ASPS prior to screening, with the exception of malignancies with a negligible risk of metastasis or death (e.g., 5-year overall survival \[OS\] rate \> 90%), such as adequately treated carcinoma in situ of the cervix, non-melanoma skin carcinoma, localized prostate cancer, ductal carcinoma in situ, or stage I uterine cancer * Treatment with systemic immunostimulatory agents (including, but not limited to, interferon and interleukin 2 \[IL-2\]) within 4 weeks or 5 half-lives of the drug (whichever is longer) prior to initiation of study treatment * Treatment with systemic immunosuppressive medications (including, but not limited to, prednisone, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor necrosis factor \[anti-TNF\] agents) within 2 weeks prior to cycle 1, day 1 or anticipation of need for systemic immunosuppressive medication during study treatment, with the following exceptions: * Patients who have received acute, low dose, systemic immunosuppressant medications or one-time pulse dose of systemic immunosuppressant medication (e.g., 48 hours of corticosteroids for a contrast allergy) are eligible after Principal Investigator confirmation has been obtained * Patients who have received mineralocorticoids (e.g., fludrocortisone), corticosteroids for chronic obstructive pulmonary disease (COPD) or asthma, or low-dose corticosteroids for orthostatic hypotension or adrenocortical insufficiency are eligible * Patients taking bisphosphonate therapy for symptomatic hypercalcemia. Use of bisphosphonate therapy for other reasons (e.g., bone metastasis or osteoporosis) is allowed * Patients with known prim