Unemployment is measured as people who are eligible and looking for work but not employed. People who have left the workforce for reasons other than getting laid off/fired (like quitting to take care of a sick family member) or people who have given up looking for work are not counted, even if they want a job. It's measured by a CPS survey of 60,000 households, and I doubt it includes homeless people (or anyone without a permanent address).

I've heard the opposition party claim the unemployment is under-counted during the Obama, Trump, and Biden years by excluding job seekers who have given up on finding employment. That's probably true, but if it is it has probably been pretty consistently under-counted for decades by both parties.