Recently, there has been significant interest in providing positioning, navigation, and timing (PNT) capability from low-Earth orbit (LEO) satellites. Providing PNT independently of medium-Earth orbit (MEO) global navigation satellite systems (GNSS) offers the potential to complement or back up these systems. LEO satellites offer several advantages over MEO satellites, including lower required transmit power and potentially lower cost-to-orbit. In parallel, there has been significant recent progress in the development of high-performance, low-power atomic clocks for space-based applications. Combining these two developments, we assess the impact of improved oscillator stability on positioning accuracy from a stand-alone LEO constellation using covariance analysis. Two techniques for position estimation are considered: pseudorange positioning, as is done using existing MEO-based GNSS, and Doppler positioning, which can leverage signals of opportunity from LEO communications satellites. We demonstrate that satellite and receiver oscillator frequency stability play a critical role in Doppler positioning accuracy and a lesser role in pseudorange positioning accuracy. We also show that, while pseudorange positioning is generally more accurate than Doppler positioning, Doppler measurements using LEO signals of opportunity can provide positioning accuracy on the order of tens of meters when highly stable clocks are used.