Per the IRS, closely held corporations have more than 50% of the value of their outstanding stock owned (directly or indirectly) by 5 or fewer individuals at any time during the last half of the tax year. And yes, some pretty huge corporations (including Hobby Lobby!) qualify as closely-held.
The sincere religious belief question was kind of elided. In cases involving real people, it's easy to establish a record of sincere belief in a trial or motion hearing -- you have the person explain the beliefs, how they came to them, etc. etc. In this new world of corporate personhood, I think it is a far more difficult question to establish what the company's religious beliefs are. The majority opinion even admits that partners in a closely held corporation can and do disagree. But for the purpose the case before it, the Court found that Hobby Lobby's family owners had established a clear record of sincere religious belief that several of the types of covered contraception were akin to abortion.