In the absence of technically mature quantum repeaters, losses in optical fibers limit the distance for ground-bound quantum key distribution (QKD). One way to overcome these losses is via optical links to satellites, which has most prominently been demonstrated in course of the Chinese-Austrian QUESS mission. Though its findings were impressive, such a large-scale project requires massive financial and time resources. We propose a 34x10x10cm nanosatellite orders of magnitude cheaper which is able to perform QKD in a trusted-node scenario, using only commercially available components. We have performed a detailed analysis of such a CubeSat mission ("Q3Sat"), finding that cost and complexity can be reduced by sending the photons from ground to satellite, i.e. using an uplink. Calculations have been done for a prepare-and-send protocol (BB84 with decoy pulses) using polarization as information carrier. We have created a preliminary design of a 3U CubeSat including a detailed size, weight and power budget and a CAD to account for the assembly of the components. Deploying a 10 cm long mirror telescope covering the small surface of the satellite leaves enough space for a polarization analysis module and housekeeping, communication and computing electronics. For one such CubeSat, we estimate the quantum secure key to be acquired between two ground stations during one year to be about 13 Mbit. A Bell test between ground and satellite would also be feasible. The uplink design allows to keep the more sensitive, computation-intensive and expensive devices on ground. The experiment proposed by us therefore poses a comparably low-threshold quantum space mission. For a two-year lifetime of the satellite, the price per kilobit would amount to about 20 Euro. In large constellations, Q3Sats could be used to establish a global quantum network, which would further lower the cost. Summarizing, our detailed design and feasibility study can be readily used as a template for global-scale quantum communication.