A laser-based, table-top instrument is constructed to perform femtosecond soft x-ray transient absorption spectroscopy. Ultrashort soft x-ray pulses produced via high-order harmonic generation of the amplified output of a femtosecond Ti:sapphire laser system are used to probe atomic core-level transient absorptions in atoms and molecules. The results provide chemically specific, time-resolved dynamics with sub-50-fs time resolution. In this setup, high-order harmonics generated in a Ne-filled capillary waveguide are refocused by a gold-coated toroidal mirror into the sample gas cell, where the soft x-ray light intersects with an optical pump pulse. The transmitted high-order harmonics are spectrally dispersed with a home-built soft x-ray spectrometer, which consists of a gold-coated toroidal mirror, a uniform-line spaced plane grating, and a soft x-ray CCD camera. The optical layout of the instrument, design of the soft x-ray spectrometer, and spatial and temporal characterization of the high-order harmonics are described. Examples of static and time-resolved photoabsorption spectra collected on this apparatus are presented.