Can I claim my unemployed girlfriend that lived with me for the entire 2024 calendar year?

Maybe.    If she lived with you the whole year and had less than $5050 of income not counting Social Security, she may be a "qualified relative" dependent for whom you get the $500 credit for other dependents.  Claiming a  GF will not make you eligible to file as Head of Household.

IRS interview to help determine who can be claimed:

https://www.irs.gov/help/ita/who-can-i-claim-as-a-dependent

Qualifying relative

• They don't have to be related to you (despite the name).
• They aren't claimed as a dependent by someone else.
• They're a U.S. citizen, resident alien, national, or a Canadian or Mexican resident.
• They aren’t filing a joint return with their spouse.
• They lived with you the entire year (exceptions apply).
• They made less than $5050 in 2024 (not counting Social Security)
• You provided more than half of their financial support.

When you add someone as a dependent, we'll ask a series of questions to make sure you can claim them. There may be other tax benefits you can get when you claim a dependent.