If I built a machine that went far away, say a buoy, would I be allowed to use ham radio frequencies to transmit telemetry data back to its base station, if I used my call sign? This is for personal research, not commercial applications.
I don't see why not as long as you followed the regulations for ID and transmission mode. There are beacons, satellites, balloons and more transmitting telemetry. Beyond the laws I would recommend following the band plans for where to transmit and pay attention to where the signals from your buoy would reach. For example, the lower end of the 70cm band isn't part of the allowed spectrum in Canada so when I go much farther North I have to pay attention and not transmit on some frequencies I can use in the US.
Make sure you follow the regulations on 'unattended operation'.
And if you are going to be floating a buoy into the sea, be aware that you may end up transmitting from the territory of another country, at which point your transmission could well be very illegal. If you ended up floating a 'message in a bottle' beacon that made its way to North Korean waters, for example, you could create an international incident!
I would say that it is probably legal, but depending on the country in which you operate it, there might be restrictions on unattended transmitters. In Switzerland, for example, only amateur radio clubs (and not "private" operators) are allowed to operate an unattended station (e.g., repeater, beacon, etc.) -- if you're member of a club, and the club is ok with that, then no problem (but of course the station has to identify itself with the club's callsign). So, although it is probably legal, you should check if there are any specific restrictions on operating unattended stations in your country.