Optical coherence tomography (OCT) with the use of soft x-rays (SXR) and extreme ultraviolet (EUV) has been recently demonstrated [Fuchs et al. Sci. Rep.6, 20658 (2016)10.1038/srep20658; Fuchs et al. Optica4, 903 (2017)10.1364/OPTICA.4.000903]. This new imaging technique, named XCT, makes it possible to obtain cross-sectional and tomographic images of objects with nanometer spatial resolution. The article presents a newly developed laboratory system for XCT using a compact laser plasma light source operating in the SXR and EUV spectral ranges. The source is based on a gas puff target containing Kr gas or a Kr/Xe gas mixture irradiated with nanosecond laser pulses from an Nd:YAG laser. The use of the gas puff target enables efficient emission of SXR and EUV radiation without generating target debris associated with laser ablation when using a solid target. The system is equipped with an ellipsoidal mirror to collect radiation from the source and focus on the imaged object. The XCT measurements are made by processing the spectrum of the radiation reflected from the object recorded with a transmission grating spectrometer equipped with an identical focusing mirror and a CCD camera. The paper presents the characterization and optimization of the new XCT system and its application to the measurements of layered nanostructures.