Really depends on your contract + country. There is no "default contract" in the EU as another commentator says[1].
You need to read your contract. They're generally fairly readable. You're looking for clauses about working for other companies and intellectual property. As the pdf I linked shows, in some countries some of these clauses end up being unenforceable, they're simply there to scare/intimidate/restrict you.
In a few of the places I worked there were prohibitions in my contract against owning or working for another company without an ok from the board/CEO/something like that. It sounds scary and unobtainable, it's definitely not.
At the first company, my friend was running a side business and asked and it was ok. At another business I already had a company when I was offered the job, I asked to keep the company as it generated a little cash each month, I was given the ok.
Generally speaking though, you're not going to have any problems. Just don't do anything using their equipment and watch out for poaching clauses in contracts if you hire people from work, they're much more likely to enforce them. The contracts I've had that actually had them were generally a year or 18 months though.
[1] Here is an example of non-compete enforcability which lists every EU company's different laws, as you can see, they're all different:
https://www.fcblegal.com/xms/files/Meritas_Guide_to_Employee...