If you connect a simple linear gain, K, in series with the 1/(s-1) block, giving a transfer function: K/(s-1), and then form a negative feedback closed loop system around this new block, the closed loop transfer function will be K/[s+(K-1)], which is stable if K>1.
Another simple block that is unstable in the open loop is an integrator, which has transfer function: 1/s. It's easy to see that this is unstable by considering a unit step input. This will result in a unit ramp output (integral of a constant, A, is ramp, At), and this goes to infinity for large t. However, closing the loop around the integrator (with or without a series gain, K) will give a stable closed loop with transfer function: K/(s+K), or 1/(s+1). As well as being stable, the closed loop system has unity steady state gain (or 'DC gain'), which is characteristic of closed loop systems that have a pure integrator in the forward path transfer function.