Inclusion Criteria:

* Patients must have histologically or cytologically confirmed adenocarcinoma of the stomach or gastroesophageal (GE) junction that has progressed after first-line treatment, or is recurrent within 6 months after receiving adjuvant therapy; patients must have had exactly one prior systemic treatment regimen; previous adjuvant (chemotherapy \[chemo\]) radiotherapy is permitted; prior chemotherapy given concurrently with radiation for radiosensitization is not considered one prior systemic regimen
* Patients must have measurable disease; computed tomography (CT) scans or magnetic resonance imaging (MRIs) used to assess measurable disease must have been completed within 28 days prior to registration; CT scans or MRIs used to assess non-measurable disease must have been completed within 42 days prior to registration; all disease must be assessed and documented on the Baseline Tumor Assessment Form (RECIST 1.1)
* Patients must not have known brain metastases
* Patients must not have received chemotherapy or radiotherapy within 3 weeks (6 weeks for nitrosoureas or mitomycin C) prior to registration
* Patient must not have received prior treatment with a phosphatidylinositol 3 (PI3), v-akt murine thymoma viral oncogene homolog 1 (AKT) or mechanistic target of rapamycin (Mtor) inhibitor for any reason
* All toxicities from prior therapy must have resolved to =< grade 1 (Common Terminology Criteria for Adverse Events \[CTCAE\] version 4.0) prior to registration
* Patients must not be receiving or planning to receive any other investigational agents
* Patients must be able to tolerate oral medications and must not have malabsorption or chronic diarrhea (CTCAE version 4.0 grade 2 or higher); administration through a feeding tube is not permitted
* Hemoglobin >= 9 g/dL
* Absolute neutrophil count (ANC) >= 1,500/mcL
* Platelets >= 100,000/mcL
* Total bilirubin =< institutional upper limit of normal (IULN)
* Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) both =< 2.5 x IULN; patients with liver metastases must have AST and ALT =< 5 x IULN
* Patients must have adequate kidney function as evidenced by at least ONE of the following:

* Serum creatinine (mg/dL) =< IULN obtained within 14 days prior to registration
* Calculated creatinine clearance > 50 ml/min; the serum creatinine value used in the calculation must have been obtained within 14 days prior to registration
* Patients must have international normalized ratio (INR) =< 1.2 unless taking therapeutic doses of warfarin; this result must be obtained within 14 days prior to registration
* Patients must have fasting blood sugar =< 150 mg/dL within 28 days prior to registration
* Patients must have hemoglobin A1C < 7% within 28 days prior to registration
* Patients must have an electrocardiogram (ECG) within 28 days prior to registration; patients must have corrected QT interval (QTcF) (by Fridericia's calculation) < 450 msec (male) or < 470 msec (female)
* Patients must have a Zubrod performance status of 0-1
* Patient must not have any of the following: a history of congenital long QT syndrome; use of concomitant medications that could prolong the QTc interval; New York Heart Association class III or IV heart failure; history of myocardial infarction within 6 months prior to registration; uncontrolled dysrhythmias; poorly controlled angina; resting heart rate =< 50 bpm (bradycardia)
* Patients must not be receiving concurrent treatment with drugs that are strong inducers or inhibitors of cytochrome P450 family 3, subfamily A, polypeptide 4 (CYP3A4); patients must be able to safely discontinue treatment with these agents for >= 2 weeks prior to beginning protocol therapy
* Patient must not have uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
* Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible
* Patient must not be pregnant or nursing; women/men of reproductive potential must have agreed to use two forms of contraception for the duration of protocol treatment and for one month after discontinuation of MK-2206; a woman is considered to be of "reproductive potential" if she has had menses at any time in the preceding 12 consecutive months; in addition to routine contraceptive methods, "effective contraception" also includes heterosexual celibacy and surgery intended to prevent pregnancy (or with a side effect of pregnancy prevention) defined as a hysterectomy, bilateral oophorectomy or bilateral tubal ligation; however, if at any time a previously celibate patient chooses to become heterosexually active during the time period for use of contraceptive measures outlined in the protocol, he/she is responsible for beginning contraceptive measures
* No other prior malignancy is allowed except for the following: adequately treated basal cell or squamous cell skin cancer, in situ cervical cancer, adequately treated stage I or II cancer from which the patient is currently in complete remission, or any other cancer from which the patient has been disease free for five years
* All patients must be informed of the investigational nature of this study and must sign and give written informed consent in accordance with institutional and federal guidelines