This is a very common debate a person trying to redecorate a room, like you, often meets.
The reasons for this dilemma are various - it might be because you are considering a complete new design of the room and wavering between these two styles, or maybe the debate is raised because you want to keep a few pieces of furniture from "old" times, but you are afraid that their style won't fit into the room's interior.
A really experienced interior designer would certainly be able to pinpoint some ideas that would solve your dilemma, but if you don't have an access to any of those people, then the following information might help you (or at least be an interesting read).
There are many instances where people have experimented putting antique pieces of furniture into the modern style room, and it worked great.
For example, if you have a very simple, linear white room done in "high class" style, an ornate antique painting would fit if you changed the color of its borders into white.
This applies even to dining sets, tables, not only paintings.
In fact, having an antique piece in your room can really create a good contrast and bring a sense of "majestic" into the ambience of your interior.
Basically, have the room done in modern style and spice it up with one or two pieces of something antique - you can't go wrong with that.
You can even combine the modern pieces with antique pieces in one item.
For example, you can have ornate legs plucked under the modern glass dining table top.
All works well, if you choose the right combinations.
Perhaps, you will need to find special places where you can order something custom made, but if you desire to have a perfect room is strong, then it is no obstacle.
Since you are buying dining sets, though, you should certainly consider the functionality of the tables and chairs - consider the amount of people who will use the dining set and determine if your room has enough space for it, and how you would like to place the table with chairs into the room.
It is not recommended to buy anything that has more than 9 pieces in the set - in these cases buying two separate sets is a better idea.
Lastly, keep in mind that modern and contemporary are two different terms, each having slightly different meanings.
Modern refers to the era of modernism (from 1880 to 1920-40), and is characterized by specific style, clean lines and well determined colors.
The main purpose of modern furniture was functionality, simplicity, comfort and obvious contours and shape.
The latter is less defined and more vague, so it is hard to characterize it.
Some shops use them interchangeably, and in fact there is not much point ruminating over the differences between the two.
Certainly, it is not recommended to buy those contemporary designs that look very unorthodox - they might look exciting, but you are risking to buy something that hasn't proven itself to be practical - they are just designer's experiments.
It might look good in designer blogs and on pictures of designer competitions, but those are just pictures.
Always go for simpler designs, if you care about functionality.
That's the reason why antique sets are usually more practical.