I am looking for an affordable plan to visit a friend, in the city of Erfurt in Germany during a saturday in October.
However, there are no flights from Paris to Erfurt Airport, and prices for a same-day flight are quite expensive for Leipzig (~500€). Why ?
Do you know of any decent alternatives ? Is my plan feasible ?
Thank you for your help. Edit : Thank you. I did not give a date because it is still uncertain yet. I am now considering sleeping in Germany
TL,DR: take the night train.
Erfurt isn't a huge city and isn't a major economic or touristic location. So there isn't much of a market for direct transportation between there and another country.
Rome2rio is a good site to look for options. Don't trust them for the details: check times and fares on specific providers' sites. But it's convenient for getting an overview.
At this distance, planes are the fastest option if there's a direct flight with a reasonably short surface journey on either side. But if you have to change planes, trains are competitive with respect to time. You can find train times on the German railways website. High-speed trains can get you there in about 6½ hours. It's too far to make a day trip practical.
There is a night train between Paris and Munich (and on to Vienna). It's a good option to arrive in Erfurt in the morning and stay until early evening.
The cheapest option (short of hitch-hiking or using your own bodily strength) is the bus. But it takes even longer than the night train and it's less comfortable. It also seems to have less convenient schedules.
Flights to and from smaller low-traffic airports are usually quite expensive because of lack of competition. But why do you want to fly to Leipzig when you are going to Erfurt? Most flights, at least the shortest ones, from Paris to Leipzig will have a transfer in Frankfurt and it is most likely just as fast to take the train from Frankfurt Airport to Erfurt (about 2:30 travel time and frequent departures) as to have an additional flight from Frankfurt to Leipzig and then go from Leipzig to Erfurt by train (about 1:00 travel time).
Considering the travel times, no matter what option you choose, I still don't understand why you want to make this a day trip.
Checking for an option on Saturday October 8th (since you don't reveal which date you are intending to travel and also seem to be slightly confused, asking about a Saturday in the text, but having searched yourself for options on a Wednesday), the first flight from Paris to Frankfurt arrives 9:40 and with about 1h time to transfer catch a train connection arriving in Erfurt 13:07.
Then, the last connection from Frankfurt back to Paris is departing 18:30 with a transfer in Amsterdam. To catch that flight, you will have to leave Erfurt 14:17 or if you risk a shorter time for checkin in Frankfurt, 14:50 may suffice. You will have something between 60 and 90 minutes to spend with your friend in Erfurt. If you book now, you can get the plane tickets for around 200€ and the train tickets for 50€. But does that make any sense at all?
Cheapest would be a Flixbus or similar but that's 11+ hours. Train is probably twice as expensive and still 8+ hours.
You can try flying to Frankfurt or Berlin and take the train from there. Same day round trips are less than $200 and maybe $70 for the train. Berlin is closer but the airport is not well connected to the city. Frankfurt airport has it's own train station so both would be about 2.5 hours one way.
Timing would be tricky for some reason it seems that the last Air France nonstop from FRA->PAR leaves already at 2:45 pm and the LH late flight is quite expensive.
This would be a lot easier if you are willing to spend a night somewhere. A Saturday night in Berlin could be a lot of fun !
