That is basically what we have built at Las Cumbres Observatory Global Telescope Network (lcogt.net). We have ~20 telescopes around the globe, all on a single network that essentially distributes a queue of observations to where and when they are most likely to be observed. The result is 24/7 coverage of the both the northern and southern hemispheres.

It's a lot harder than you'd think. There is no such thing as an off the shelf professional grade telescope, so we employ a large engineering team to actually build them. Scopes and their instruments are complex and made of many independent systems and require a TON of software to run without an human operator (and even with). Scheduling observations is another challenge, an np-complete one actually. Then there's the problem of analyzing and storing gigabytes of data a day.

As a member of the relatively small software team, we have a TON to do. But it's a really cool problem to work on!