Welcome to the Forum!
Yes you can do what you want. Bigger machines cost more (typically) but can do larger work pieces. You can index parts to re-position as you mention but the machine needs to have clearances to be able to do this. It is often more work but for hobby use this may not be an issue.
Might be worth searching and reading various build posts on this forum to get some background. You will need to identify what sort of accuracy you want to achieve and decide how much budget you have. Add the cost of tooling to your budget as well. Machines can be built from different materials including various woods, metals (aluminium, steel) and combinations. Materials have their different properties, strengths and weaknesses.
Determining what is right for you goes back to what you want to do, how much precision, performance and other factors including cost.
Building machines from scratch can take a lot of effort - especially to get accuracy. Kits are popular, some better than others. Chinese machines can be good value for money but be prepared to spend extra time and effort making them "just right" for your needs. Sometimes they lack quality of finish, attention to detail or are just badly put together. If you go the Chinese path, be absolutely sure the supplier understand what you want. They are good at saying they understand what you ask for up front but sometimes do not deliver what you think you have both agreed on. Ask for pictures etc.
Second hand machines can be good value sometimes even if they need some maintenance or repairs - look for auctions or folks seeking bigger and better machines.
Feel free to search the forum or ask questions - there are a lot of smart and experienced folks here.
Allow for growth (eg: 4th axis as mentioned) as you learn.
Good hunting :-)