Radial velocity observations of stars are entering the sub-m/s domain revealing fundamental barriers for Doppler precision experiments. Observations of the Sun as a star can easily overcome the m/s photon limit but face other obstacles. We introduce the G\"ottingen Solar Radial Velocity Project with the goal to obtain high precision (cm/s) radial velocity measurements of the Sun as a star with a Fourier Transform Spectrograph. In this first paper, we present the project and first results. The photon limit of our 2 min observations is at the 2 cm/s level but currently limited by strong instrumental systematics. A drift of a few m/s per h is visible in all observing days probably caused by vignetting of the solar disk in our fiber coupled setup, and imperfections of our guiding system adds further offsets in our data. Binning the data into 30 min groups shows m/s stability after correcting for a daily and linear instrumental trend. Our results show the potential of Sun-as-a-star radial velocity measurements that can possibly be achieved after a substantial upgrade of our spectrograph coupling strategy. Sun-as-a-star observations can provide crucial empirical information about the radial velocity signal of convective motion and stellar activity, and on the wavelength dependence of radial velocity signals caused by stellar line profile variations.
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