Space-time shift keying-aided orthogonal time frequency space modulation-based multiple access (STSK-OTFS-MA) is proposed for reliable uplink transmission in high-Doppler scenarios. As a beneficial feature of our STSK-OTFS-MA system, extra information bits are mapped onto the indices of the active dispersion matrices, which allows the system to enjoy the joint benefits of both STSK and OTFS signalling. Due to the fact that both the time-, space- and DD-domain degrees of freedom are jointly exploited, our STSK-OTFS-MA achieves increased diversity and coding gains. To mitigate the potentially excessive detection complexity, the sparse structure of the equivalent transmitted symbol vector is exploited, resulting in a pair of low-complexity near-maximum likelihood (ML) multiuser detection algorithms. Explicitly, we conceive a progressive residual check-based greedy detector (PRCGD) and an iterative reduced-space check-based detector (IRCD). Then, we derive both the unconditional single-user pairwise error probability (SU-UPEP) and a tight bit error ratio (BER) union-bound for our single-user STSK-OTFS-MA system employing the ML detector. Furthermore, the discrete-input continuous-output memoryless channel (DCMC) capacity of the proposed system is derived. The optimal dispersion matrices (DMs) are designed based on the maximum attainable diversity and coding gain metrics. Finally, it is demonstrated that our STSK-OTFS-MA system achieves both a lower BER and a higher DCMC capacity than its conventional spatial modulation (SM) {and its orthogonal frequency-division multiplexing (OFDM) counterparts. As a benefit, the proposed system strikes a compelling BER vs. system complexity as well as BER vs. detection complexity trade-offs.