Inclusion Criteria:

* Men and women \>= 18 years of age
* Patients will need to have one of the following clinical scenarios:

* Previously untreated follicular lymphoma grade 1-3a with low tumor burden by Groupe d'Etude des Lymphomes Folliculaires (GELF) criteria
* Previously untreated follicular lymphoma grade 1-3a with high tumor burden by GELF criteria but who are unable or unwilling to receive standard front-line treatment approaches
* Previously untreated marginal zone lymphoma, lymphoplasmacytic lymphoma, or any other indolent B-cell lymphoproliferative disorder with low tumor burden by GELF criteria or who are unable/unwilling to receive more intensive front-line treatment
* Previously untreated mantle cell lymphoma who would otherwise be appropriate candidates for watchful waiting OR who have symptomatic disease but are not candidates for or decline standard induction approaches
* Patients with previously untreated low tumor burden FL (criterion above) must have measurable and/or assessable disease defined as at least one involved lymph node or extranodal disease site that measures \>= 1.5cm in greatest diameter
* Patients who meet inclusion criteria above are eligible as long as they meet one of the following criteria for measurable/assessable disease:

* At least one involved lymph node or extranodal disease site measuring \> 1.5cm in greatest diameter
* Pathologically-confirmed bone marrow or peripheral blood involvement that can be reassessed for response
* Pathologically confirmed splenic or extranodal involvement with at least one known site of disease remaining after diagnostic biopsy that can be reassessed (i.e., patients with splenic marginal zone lymphoma who complete splenectomy and have no other detectable disease would not be eligible)
* Eastern Cooperative Oncology Group (ECOG) performance status of 0-2
* Woman of childbearing potential (WOCBP) and men enrolled on this protocol must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry for the duration of study participation, and for at least 2 days after the last dose of acalabrutinib or 18 months after the last dose of obinutuzumab, whichever is longer. 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
* Women of childbearing potential must have a negative serum or urine pregnancy test prior to starting therapy
* Willing and able to participate in all required evaluations and procedures in this study protocol
* Ability to understand the purpose and risks of the study and provide signed and dated informed consent and authorization to use protected health information Ability to understand the purpose and risks of the study and provide signed and dated informed consent and authorization to use protected health information

Exclusion Criteria:

* The presence or history of histologically transformed or co-existing high-grade or aggressive non-Hodgkin lymphoma
* Confirmed active or prior central nervous system disease
* Prior receipt of lymphoma-directed therapy or prior antibody-based therapy (except for anti-microbial therapy for infection-associated marginal zone lymphoma such as hepatitis C or H pylori)

* A short course of steroids is permitted for patients aside from those in the low tumor burden FL cohort. This course may be no more than 14 days and steroids must be discontinued (or tapered to =\< 10mg prednisone or equivalent) no later than 3 days after initiation of study treatment. Patients in the low tumor burden FL cohort may not receive corticosteroids as an anti-lymphoma therapy at any time before starting treatment
* Prior malignancy (or any other malignancy requiring active treatment), except for adequately treated basal cell or squamous cell skin cancer, in situ cervical cancer, or other cancer from which the subject has been disease free for \>= 2 years or which will not limit survival to \< 5 years
* Clinically significant cardiovascular disease such as symptomatic ventricular arrhythmias, congestive heart failure, or myocardial infarction within 6 months of screening, or any class 3 or 4 cardiac disease as defined by the New York Heart Association Functional Classification. Note: Subjects with controlled, asymptomatic atrial fibrillation can enroll on study if deemed appropriate by the investigator
* Has difficulty with or is unable to swallow oral medication, or has significant gastrointestinal disease that would limit absorption of oral medication
* Known history of human immunodeficiency (HIV) or any active significant infection (e.g., bacterial, viral, or fungal) within 14 days of cycle 1. Patients with uncomplicated viral or bacterial infections that are being managed with oral antibiotics and/or supportive care alone are eligible
* Known history of hypersensitivity or anaphylaxis to study drug(s) including active product or excipient components
* Active bleeding or history of bleeding diathesis (e.g., hemophilia or von Willebrand disease)
* Uncontrolled AIHA (autoimmune hemolytic anemia) or ITP (idiopathic thrombocytopenic purpura)
* Presence of a gastrointestinal ulcer diagnosed by endoscopy within 3 months before screening
* Requires treatment with a strong cytochrome P450 3A4 (CYP3A4) inhibitor/inducer
* Requires or receiving anticoagulation with warfarin or equivalent vitamin K antagonists
* Requires treatment with proton pump inhibitors (e.g, omeprazole, esomeprazole, lansoprazole, dexlansoprazole, rabeprazole, or pantoprazole). Note: Subjects receiving proton pump inhibitors who switch to H2-receptor antagonists or antacids are eligible for enrollment to this study
* History of significant cerebrovascular disease/event, including stroke or intracranial hemorrhage, within 6 months before the first dose of study drug. Patients with a transient ischemic attack which has resolved and for which there are no ongoing symptoms are eligible
* Major surgical procedure within 28 days of first dose of study drug (not including a diagnostic procedure to make the lymphoma diagnosis). Note: If a subject had major surgery, they must have recovered adequately from any toxicity and/or complications from the intervention before the first dose of study drug
* Hepatitis B or C serologic status: subjects who are hepatitis B core antibody (anti-HBc) positive and who are hepatitis B surface antigen (HBsAg) negative will need to have a negative polymerase chain reaction (PCR) and must be willing to undergo deoxyribonucleic acid (DNA) PCR testing during the study to be eligible. Those who are HBsAg positive or hepatitis B PCR positive will be excluded. Subjects who are hepatitis C antibody positive will need to have a negative PCR result to be eligible and have completed appropriate anti-viral treatment. Those who are hepatitis C PCR positive will be excluded. Anti-viral therapy for patients with hepatitis-C associated marginal zone lymphoma will not be considered a prior anti-lymphoma treatment
* Absolute neutrophil count (ANC) \< 1,000/mcL
* Platelet count \< 50,000/mcL (Unless felt to be related to underlying disease)
* Total bilirubin \>= 1.5 x the upper limit of normal (ULN). Isolated bilirubin \> 1.5 x ULN is permitted if the direct proportion is \< 35%
* Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) \> 2.5 x ULN
* Creatinine clearance =\< 40 mL/min/1.73m\^2
* Breastfeeding or pregnant
* Concurrent participation in another therapeutic clinical trial