Can a Navy sailor, that is a single E-5 with no dependents, receive
BAH (basic allowance for housing) and get paid for off-base rental
expenses while stationed in Yokosuka, Japan?

Off base housing at the Yokosuka Welcome Center information can be
found at http://housing.cnfj.navy.mil/

The abilty for an unaccompanied service member to live on the local
economy is based on occupancy of the on base quarters. It's normal to
have E1 - E4 stay on base, if unaccompanied. E5 is the gray area, with
some commands allowing/encouraging them to live off base and others
requiringthem to stay on base. E6 and above are usually allowed to
move off base with no problems. Any chance you could make rank before
arriving? :-)

I visited a friend of mine at Yokosuka several years ago, and at the
time, it was possible for an E5 to move off base.

You should be able to contact the housing office at the link above for
more up-to-date info.