70mm diameter makes something like this mini-lathe too small. You could do it on one of those, but I can say from past experience that it would be unpleasant!

Next size up would be something like a Myford lathe, but they tend to be overpriced and are still a bit of toy really. What you really want is something around the size of a Colchester student (like I have), or a Harrison M250/300. Colchester student's go for around £500 on eBay, so well within your budget if converted to CNC, which wouldn't be necessary for the part pictured. The issue is they are very heavy, so unless you can get to your garage/shed from the road, it's likely to be difficult to get the lathe in place. Any decent lathe will be very heavy - you could probably judge how good a lathe is by weighing it.