The Lion Inn certainly stilldoes Bed and Breakfast, the prices are on their website. The real issue is that they are moving more towards guaranteed bookings from the Middlesbrough toffs rather than single bed walkers who are not always turning up. I passed last Sunday and their car park was reasonably full. If you are arriving midweek there should be little problem booking from your previous stopping point, weekends could be a tad more problematic if they won't accept a booking.
An alternative is to catch the Moorsbus service that stops outside the pub. From the pub you can catch a bus direct into Hutton (very expensive accommodation) or Castleton (far cheaper). If you continue to Ralph Cross can also travel into Rosedale Abbey and there is another service into Castleton (that doesn't pass the Lion) from here. There is also the bunkhouse at Sinnington.
The timetable is usually out well before Easter, usually about February, but the timings may be awkward. This year a bus left Castleton for Ralphs just before 10. They run weekends most of the Summer (with a limited midweek service) with a service everyday during the mains school holidays.
You can also catch a train from Castleton to the larger villages EG the 9.15 (approx) train from Glaisdale connects with the 10am (approx) bus.
Failing that, some B and B in Castleton will come and pick you up and drop you back the next day. Contact them and ask.