Finding a host that can provide enough server space is probably the easier part. Although, be cautious of companies that suggest “unlimited space” because there are usually limits.
When not bogged down by too many poorly designed components, Joomla! is actually a lightweight CMS that does well on busy sites. We actually had one of our boutique hosts tell us that during an extremely busy time with one of our Joomla! sites it way out-performed what any WordPress site could have done in terms of utilization of bandwidth and server resources. So you’ve got a good CMS for a big site.
Look for plenty of space. Look for Linux hosting. Make sure they are up to date with PHP/MySQL versions. Take note of bandwidth limitations.
As for individual hosts to choose, this is always tough to determine. I’ve been doing this for 15 years and I’ve seen good hosts turn bad when they oversell space. And many won’t tell you until after you’re up and running that your only way to solve your issue is to go to their much more expensive Dedicated Servers, which isn’t always correct. Maybe it’s just finding a better host.
But large sites usually end up on VPS or Dedicated Servers in the end. Just know, on a Dedicated Server, you may be left managing it yourself or paying a fee for service.
At the Joomla! conference, I saw companies like Rochen and SiteGround as sponsors. They are very Joomla! friendly hosts. I don’t happen to use either one, but that is because they don’t my particular needs as a reseller. I wouldn’t point you away from them. They may be a good fit for you. A smaller boutique hosting company might be an even better choice, if you want someone who can configure your server the way you want. But you’ll likely pay more. In addition, take into consideration SSL Certificates and the needs of PCI Compliance if you’re going to be selling anything.
I’d recommend talking to the host’s sales team. Raise any concerns and see if they have a money-back guarantee. And don’t commit to more than a year, until you know the host is a good fit.
I’m sure there are plenty of other suggestions out there, but hopefully this is food for thought and of some help.