Can a non-periodic function have a Fourier series? Consider two periodic functions. Assume their sum is not periodic.
The periodic functions can be represented by a Fourier series. If you add up the Fourier series, you get a series that represents their sum. But their sum is not periodic, yet you have described it using a Fourier series.
I thought that non-periodic functions can't be represented by a Fourier series. Why isn't this a contradiction?

A Fourier series means the amplitude of the different harmonics, who are an integer multiple of a base frequency.
It is easy to see, that this base frequency simply doesn't exist in your case.
Although a Fourier transform of a such function of course exist, which is trivially 
$$F(s)=\delta(t-\omega_1)+\delta(t-\omega_2)$$
