Despite the terrible name, I was blown away!
Stunning interior and fantastic staff. I was slightly terrified on my first visit of breaking something (be very careful with your bags knocking things), but definitely worth looking around. Don't be afraid to ask one of the staff for a tour either (if they're quiet in there), it really is fascinating to get their insight, especially the collections and rarities they have there.
Also, downstairs there are wine sampling machines that allow you to taste various wines in 25ml samples (above that becomes a licensing issue). You ask for a smart card (like an Oyster card), put some money on it, then help yourself! There difference here is that some wines are just a pound or so, whereas others are £30+ for your 25ml sample. These are wines that you never even see available by the glass elsewhere, so can be a real treat to taste. I've been a couple of times just to do that with friends.
Highly recommended! (if you hadn't guessed!)
A few places that you also might like that I can think of are:
Arbutus in Soho - modern European Michelin star restaurant. Has a small counter of maybe 8 seats at the bar. I've always found it to be very good. Pre-theatre menu is great value.
Pizza East in Shoreditch. Not really a bar, but you can sit at the counter and watch the chefs making the pizza. They're busy, but I've had some fun banter with the chefs and other diners sitting there. Good, thin, Italian style pizza... my American friends have always liked it. Fun, buzzing dining room.
Riding House Cafe near Oxford Circus is a decent brunch/lunch spot (not had dinner there). There's a large communal table that often gets some chatter going. Think Schiller's or Essex in New York's Lower East Side. The ground floor at Smith's of Smithfield in Farringdon is also a similar proposition with some big communal tables.
As mentioned, Opera Tavern in Covent Garden has dining at the bar too. Very good Spanish tapas . I still have dreams about the scallops I ate there once.