It depends on what you want to use the gym for ie do you want to do aerobics courses as well as the machines?
20€ a month sounds quite good to me I paid 45€ in my old gym but I guess Frankfurt prices are higher.
They should give you a free trial of either an aerobics course or a session in the gym or maybe both.
There are some gyms that charge an upfront fee. The question is do you think you will stick at it for 15months? Also does that mean no membership fee for the first 15 months if you pay the 300€. Are there extra charges such as a yearly club membership?
If it was me and I had 300€ to spare I would probably go for it - after checking the small print of course.
I now go to the Turngemeinde Bornheim and it costs me 9€ a month just for aerobics it is not a luxury fitness studio but I think it suits my needs.
I would also consider how close it is to home if you can walk or if not are there parking spaces there as I used to find half the hassle was getting there!!
Hope this has helped a little