Social Security requires a medically determinable physical or mental impairment as the basis for a finding of disability. The "medically determinable" language is part of the definition of disability in the Social Security Act itself. See 42 U.S.C. 423(d)(1)(A).
A physical or mental impairment must be established by medical evidence consisting of signs, symptoms, and laboratory findings, not only by your statement of symptoms. Please see 20 C.F.R. 404.1508. Signs, symptoms and laboratory findings are defined by regulation. Also, Social Security Ruling 96-4p discusses this requirement, and is a good starting place for research on the issue.
The point of the requirement for a medically determinable impairment (MDI) is to ensure there is a valid basis for your claim, and for the functional limitations you are asserting.
Some illnesses are problematic in the MDI department, however. Social Security has had to clarify that the MDI requirement is met for certain illnesses. For example, Social Security has declared that chronic fatigue syndrome, post-polio sydrome and fibromyalgia all satisfy the requirement for a medically determinable impairment. As a result, these impairments can medically support a determination of Social Security disability.