In this paper, we introduce the concept of graph shift-invariant space (GSIS) on an undirected finite graph, which is the linear space of graph signals being invariant under graph shifts, and we study its bandlimiting, kernel reproducing and sampling properties. Graph bandlimited spaces have been widely applied where large datasets on networks need to be handled efficiently. In this paper, we show that every GSIS is a bandlimited space, and every bandlimited space is a principal GSIS. Functions in a reproducing kernel Hilbert space with shift-invariant kernel could be learnt with significantly low computational cost. In this paper, we demonstrate that every GSIS is a reproducing kernel Hilbert space with a shift-invariant kernel. Based on the nested Krylov structure of GSISs in the spatial domain, we propose a novel sampling and reconstruction algorithm with finite steps, with its performance tested for well-localized signals on circulant graphs and flight delay dataset of the 50 busiest airports in the USA.