Inclusion Criteria:  Participants must be able to understand and willing to sign a written informed consent document. Participant must have signed and dated an IRB/IEC approved written informed consent form in accordance with regulatory and institutional guidelines. This must be obtained before the performance of any protocol-related procedures that are not part of normal subject care. Participant must be willing and able to comply with scheduled visits, treatment schedule, laboratory tests, and other requirements of the study. Participants must be at least 18 years old on day of signing informed consent. Participants must have a ECOG Performance Status 0-1 (see Appendix A). Life expectancy of > 3 months (in the opinion of the investigator). Participants with recurrent or refractory (R/R)R/R must have histologically confirmed DLBCL CNS lymphoma (from brain biopsy, CSF or vitreous biopsy, and includes PCNSL and SCNSL) for Phase I; R/R histologically confirmed DLBCL PCNSL (from brain biopsy only) for Phase II. Participants should have evidence of R/R disease on MRI or on CSF cytology. Participants must have received at least 1 line of CNS-directed prior therapy. There is no maximum limit on the number of prior therapies. Confirmation of availability of sufficient tissue from brain biopsy for correlative studies is required prior to enrollment (for phase II only); these samples must be sent to the DFCI Coordinating Center within 60 days of registration.  The following amount of archived tissue is required: At least 10 but up to 20 unstained formalin-fixed, paraffin-embedded (FFPE) slides. Histologically confirmed tissue will be required from the time of relapse or at the time of initial surgery.  Participants must have recovered to ≤ grade 1 or pre-treatment baseline from clinically significant toxic effects of prior therapy. Participants must be able to undergo MRI.  Participants must demonstrate adequate as defined below (all screening labs should be performed within 14 days of registration but before 1st dose of study drug):  Hematology  White Blood Count (WBC) ≥ 2 K/µL Platelet count ≥ 100 K/µL Absolute Neutrophil Count ≥ 1.5 K/µL Hemoglobin > 9.0 g/dL or ≥ 5.6 mmol/L (Criteria must be met without erythropoietin dependency and without packed red blood cell (pRBC) transfusion within last 2 weeks) Serum creatinine ≤1.5 x institutional ULN OR Measured or calculated creatinine clearance ≥30 mL/min for participant with creatinine levels >1.5 × institutional ULN (Creatinine clearance should be calculated per institutional standard) Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 2.5 x ULN (≤5 × ULN for participants with liver metastases) Total bilirubin (TBILI) ≤ 1.5 x institutional ULN (except subjects with Gilbert Syndrome who must have a total bilirubin level of < 3.0 x institutional ULN) OR Direct bilirubin ≤ULN for participants with total bilirubin levels >1.5 × ULN)  Woman of childbearing potential (WOCBP) who are sexually active must use highly effective methods of contraception during treatment and for 2 days after the last dose of acalabrutinib. For male subjects with a pregnant or non-pregnant WOCBP partner, no contraception measures are required. Highly effective methods of contraception include:  Combined (estrogen and progestogen containing) hormonal contraception associated with inhibition of ovulation, which may be oral, intravaginal, or transdermal Progestogen-only hormonal contraception associated with inhibition of ovulation, which may be oral, injectable, or implantable Intrauterine device (IUD) or intrauterine hormone-releasing system (IUS) Bilateral tubal occlusion Vasectomy of a female subject's male partner (with medical assessment and confirmation of vasectomy surgical success) Sexual abstinence (only if refraining from heterosexual intercourse during the entire period of risk associated with the study treatments)  Exclusion Criteria:  Participants unable to undergo MRI brain. Participants with > Grade 2 intracranial hemorrhage. Participants with active systemic disease. Participants with uncontrolled intercurrent illness. Participants with prior exposure to BTK inhibitors 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 ≥ 3 years. Participants who have received prior systemic anti-cancer therapy including investigational agents or radiotherapy within 4 weeks prior to dosing. OR 5 half-lives, whichever is shorter Note: Participants must have recovered from all AEs due to previous therapies to ≤Grade 1 or baseline. Participants with ≤Grade 2 neuropathy may be eligible. Clinically significant cardiovascular disease such as symptomatic 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 or corrected QT interval (QTc) > 480 msec at screening. Note: Subjects with controlled, asymptomatic atrial fibrillation can enroll on study. 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 infection with HIV, prior history of PML or any active significant infection (eg, bacterial, viral, or fungal). Known history of hypersensitivity or anaphylaxis to acalabrutinib including active product or excipient components. Active bleeding or history of bleeding diathesis (eg, hemophilia or von Willebrand disease). Uncontrolled AIHA (autoimmune hemolytic anemia) or ITP (idiopathic thrombocytopenic purpura). Requires treatment with a strong cytochrome P450 3A4 (CYP3A4) inhibitor/inducer. Requires or receiving anticoagulation with warfarin or equivalent vitamin K antagonists. Prothrombin time (PT)/INR or aPTT (in the absence of lupus anticoagulant) >2x ULN. Requires treatment with proton pump inhibitors (eg, 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. Major surgical procedure within 28 days of first dose of study drug. 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 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. Those who are hepatitis C PCR positive will be excluded. Breast feeding or pregnant Concurrent participation in another therapeutic trial. Liver cirrhosis categorized at Child Pugh Score C. Uncontrolled hypertension despite optimal medical management.