Most internet start-ups these days don't manage their own hardware, unless they've got several hundred million dollars in funding and some fairly specialist needs. Pretty much everyone is working just fine on rented virtual machines, dedicated servers, and utility-pricing-model 'cloud' services. This doesn't mean that they don't need a sysadmin, or 'ops' team. However, just because you don't manage the physical hardware doesn't mean that the network will somehow manage itself. Unless you tie yourself to an entirely managed platform like Heroku or Google App Engine, you still need to pick an OS, maintain it, and make all the same decisions that operations people have always made.
(And if you are tying yourself to Heroku, fair enough, but you're going to end up paying for their free customers' resource utilisation and sysadmin staff costs. PaaS is great if you're small and probably won't grow, but long-term it's a great way to waste money.)
The only difference between the operations role with hardware and the operations role without hardware is that your operations team don't need to be in the same location, because they're not responsible for maintaining the hardware. This is where I come in - I reckon that American companies, especially trendy internet start-ups, would do well to hire ops staff from outside America, particularly people based in the UK but outside of London.
I'll be Terribly Subtle and use myself as an example. I've around seven years of Linux server management experience (including actual hardware, like the ancient fossil that I am), and I've been using Linux in one way or another since 1999, when it was actually pretty hard to get Linux working on a Mac. (First attempt: SuSE running under VirtualPC, back when it was an x86 emulator and before Microsoft bought it, second attempt: Yellow Dog Linux dual-booting on my B&W G3, that one was actually usable.) I've managed hundreds of VMs and physical machines without using configuration management tools, and I've managed dozens using configuration management. I know how to scale Ruby applications and how to write Chef recipes. I've been in a 'DevOps' role, and understand the value of DevOps, especially for small teams. Since you can't get pagers any more, I have an on-call phone with a week-long battery life instead. I think I'm a fairly typical ops type; I even have a substantial beard.
I also live just outside Edinburgh, and the average Bay Area Unix Sysadmin earns more than twice as much as me. Which is to say, if a Bay Area startup can put up with an eight hour time difference (which can be a benefit as well as an obstacle) they can have an ops person for far less than they think. (Similar maths applies to contracting, so if you want a specialist to roll up and help set up your infrastructure automation it could be cheaper than trying to muddy along without it.)
There's a stigma attached to 'outsourcing' or hiring overseas staff, but I think that in this case at least it's faulty: this isn't bulk-purchasing mediocre workers because they're cheaper, this is targeted hiring of competent and skilled workers because they provide a different set of benefits. There's no language barrier, and the time zone difference can actually be a benefit for ops staff: what's 'out of hours' to a team based in San Francisco is potentially 'standard business hours' to the UK. This gives easier emergency coverage (server implode at 3am PST? No need to wake anyone up, that's right in the middle of my working day) and allows for less stressful scheduled out of hours work (does anyone really think that those DB migrations at midnight aren't harder to do than they would be during the day?).
It's possible to take this beyond just hiring a UK sysadmin. Distributed teams can be more flexible, adaptable, and agile. With a team spread across different countries and continents you can benefit from a range of different viewpoints and can be more aware of different cultural concepts. This can help you avoid traps caused by the entire team operating on the same set of cultural assumptions. Diversity makes for stronger teams and more robust products! But for a company that's not ready to go 'fully distributed', I think that operations staff are a good place to test the waters.
And if that's convinced you to try it out, why not give me a shout!