Research conducted in the framework of the Jagiellonian-PET (J-PET) project aims to develop a cost-effective total-body positron emission tomography scanner. As a first step on the way to construct a full-scale J-PET tomograph from long strips of plastic scintillators, a 24-strip prototype was built and tested. The prototype consists of detection modules arranged axially forming a cylindrical diagnostic chamber with the inner diameter of 360 mm and the axial field-of-view of 300 mm. Promising perspectives for a low-cost construction of a total-body PET scanner are opened due to an axial arrangement of strips of plastic scintillators, which have a small light attenuation, superior timing properties, and the possibility of cost-effective increase of the axial field-of-view. The presented prototype comprises dedicated solely digital front-end electronic circuits and a triggerless data acquisition system which required the development of new calibration methods including time, thresholds, and gain synchronization. The system and elaborated calibration methods, including first results of the 24-module J-PET prototype, are presented and discussed. The achieved coincidence resolving time equals to ${\mathrm {CRT}}=490\pm 9$ ps. This value can be translated to the position reconstruction accuracy $\sigma (\Delta l) =18$ mm, which is fairly position independent.