Mass exchange and mass loss in close binaries can significantly affect their evolution, but a complete self-consistent theory of these processes is still to be developed. Processes such as radiative shielding due to a hot-spot region, or a hydrodynamical interaction of different parts of the gas stream have been studied previously. In order to test the respective predictions, it is necessary to carry out detailed observations of binaries undergoing the largescale mass exchange, especially for those that are in the rapid transfer phase. {beta} Lyr A is an archetype of such a system, having a long and rich observational history. Our goal for this first study is to quantitatively estimate the geometry and physical properties of the optically thick components, namely the Roche-lobe filling mass-losing star, and the accretion disk surrounding the mass-gaining star of {beta} Lyr A. A series of continuum visible and NIR spectro-interferometric observations by the NPOI, CHARA/MIRC and VEGA instruments covering the whole orbit of {beta} Lyr A acquired during a two-week campaign in 2013 were complemented with UBVR photometric observations acquired during a three-year monitoring of the system. We included NUV and FUV observations from OAO A-2, IUE, and Voyager satellites. All these observations were compared to a complex model of the system. It is based on the simple LTE radiative transfer code SHELLSPEC, which was substantially extended to compute all interferometric observables and to perform both global and local optimization of system parameters. Several shapes of the accretion disk were successfully tested - slab, wedge, and a disk with an exponential vertical profile - and the following properties were consistently found: the radius of the outer rim is 30.0+/-1.0R_{sun}, the semithickness of the disk 6.5+/-1.0R{sun}_, and the binary orbital inclination i=93.5+/-1.0deg. The temperature profile is a power-law or a steady-disk in case of the wedge geometry. The properties of the accretion disk indicate that it cannot be in a vertical hydrostatic equilibrium, which is in accord with the ongoing mass transfer. The hot spot was also detected in the continuum but is interpreted as a hotter part of the accretion disk illuminated by the donor. As a by-product, accurate kinematic and radiative properties of Lyr B were determined.