Can a tenant be asked to leave with only 5 days notice?

Yes, what you have is a lease, it is called a month to month most likely in your situation, even though it is not written.  When you are there and pay periodically, you must be given the period's notice to move. So for a thirty day, in some states you must be given thirty days.  See this link from freeadvice.com, http://real-estate-law.freeadvice.com/landlord_tenant/year_lease.htm

You can sue for any moving costs incurred and additional costs for finding new housing.  Florida law requires notice to the tenant be in writing, even if the lease was oral.  If the eviction was due to your fault, based on the severity of the issues, the Florida landlord may only have to give you 7 days written notice but no less than 7 days.

In Florida (in an oral or non-term lease), you must be given 7 days notice for weekly rent, 15 days for monthly and 30 days for quarterly.