I've usually gotten the tube or driven to get to the ENO, so can't directly answer your question, but it's very close to the St. Martin's Lane hotel, and presumably, that will be swarming with cabs.
I'm also not sure about the bag storage issue (not much help, am I?), but their telephone number, from the website, is either 020 7632 8300 or 0871 911 0200.
tb