France to Doha without flying

First step, Paris-Piraeus is relatively easy: there's a bus to Beograd, then the train to Thessaloniki, Athens and finally the port.

Then you could use freighter travel from Piraeus, Greece to Jeddah, Saudi Arabia. Then Saptco bus will take you to Doha, Qatar on three buses.

Alternatively, you could use another freighter to start from Fos sur Mer, next to Marseille, France and arrive to Jeddah two weeks later. Although this ship takes you to Jebel another week later which is physically closer but you are stuck without public transit options.

You are short on options. While it seems there is a ferry from Suez to Duba, Saudi Arabia you can't get to Egypt: Are there any ferries from Europe to Egypt? . You can't go around this by perhaps ferrying to Israel, crossing into Taba and taking the bus to Suez: there's no ferry to Israel any more and I am not sure East Delta runs those buses any more and I am almost sure it'd be a very dangerous journey for a white European to cross the Sinai on a bus if it existed. Otherwise, it's good :)

Nor you can get there from Iran: the Bushehr - Doha ferry is currently not operational, according to http://caravanistan.com/transport/persian-gulf-ferry/ and also http://www.valfajr.ir/121/index.aspx (use Google Translate) does not have it and supposedly they ran this ferry.

Edit: I forgot an important detail: Saudi Arabia and its visa situation. Unless you are muslim or have family in Saudi Arabia you have almost zero chance of gaining entry into that country. So either take the second freighter to Jebel Ali or go around and take the ferry from Bandar Lengeh, Iran to Dubai. Both ways you end up near or in Dubai and you will need to rent a car or hire someone to drive you to Doha -- but that again takes you through Saudi Arabia. However, for this section I have found some expats managed to get a visa on the border. YMMV.

You might need to get in contact with Qatar National Navigation & Transport Co. Ltd and ask them whether they know of any freighter starting in Europe taking passengers which will go to Doha. There doesn't seem to be a regular option. Sorry!

On arrival visas are only available at the airport; and if you want to drive to Doha you'll have to get a transit visa for Saudi Arabia (and these are only issued to those already possessing visas for their destination country).

You also cannot rent a car and drive it to Doha from UAE, you'd have to buy a vehicle (or bring your own). If so, make sure also have a CPD with you, as this is required for UAE and Qatar for cars that are not registered within the GCC.

In short - driving is out; unless you are able to get a visa in advance to Doha; because this is a requirement to get a transit visa through Saudi Arabia.

Your next option is to take ferry; and there are plenty of options for that.