I've had the same issue (there also are 2 threads on steam forums for this bug).

I'm pretty sure the issue here is that you can't start the space program (or load a saved one) when your graphics settings are too low. It's working with the low preset but manually setting it even lower results in an unplayable game.

Here's my story:

My system is a Samsung Series 7 laptop 740U3E (win 8) with an AMD Radeon HD8570M graphics card. After installation I first tried the manned scenarios and discovered that my graphics card lacks severely in performance for this game. Therefore I set the resolution to 1280x720 and the settings to low. I was still not satisfied with the frame rate and manually set the graphics options to the lowest (bilinear filtering, all bars to the leftmost), then it was playable.

However, on starting the space program I successfully completed the prologue mission but on the following loading screen the game just froze.

After trying a lot of things (freeze on alt-tab, compatiblity settings, steam settings for cloud and steam overlay, switching to the integrated IGP4000 graphics chip) I thought maybe it's just this first launch of the career mode that's not working and I could use another PC to skip that step on my laptop and transfer the savegame.

I installed Steam and Take On Mars on a desktop pc (win 7) with a GeForce 8600GT (far worse than the laptop), set the same settings (720p, manually all options to the lowest possible) and got the same freeze on the loading screen after the prologue mission.

Curious as to how that could happen on two so different platforms and after reading a lot of general advice on how to fix these kind of bugs I launched the game on the laptop but chose 'Set to default settings' on the launch app and started the game and voila - I could successfully start the space program part of the game.

I still had a very poor performance on graphics (due to default settings) and tried to lower the settings again. After an hour or so of testing I came to the conclusion that the game works at preset low but manually setting the graphics settings even lower results in the freeze on the loading screen. The resolution appears to be not relevant to this bug (although I only tried on 720p).

I didn't test this further but reproducibility seems to be always on all platforms if graphics settings are manually set as low as possible.

I'll be watching this issue further so if there are any questions just add a note to the issue.