1. A method in a computer system for modernizing an existing menu-based legacy application, comprising: by a client application executing within a Web browser executed by the computer system, receiving an indication of a user defined key sequence; associating the user defined key sequence with an executing task of the existing menu-based legacy application executing on a host system; and facilitating access to the executing task by displaying screen output of the executing task whenever the associated user defined key sequence is entered, wherein the computer system is a client computing system that interacts with a remote computer system that hosts the existing menu-based legacy application, wherein the menu-based legacy application presents screen output via a series of hierarchical menus, each menu represented as a character stream for presentation line-by-line on a distinct terminal screen without pixel addressing, the distinct terminal screen being one of a plurality of distinct terminal screens, and wherein facilitating access to the executing task includes displaying the screen output of the executing task in one user action of entering the defined key sequence instead of by navigating the series of hierarchical menus displayed line-by-line on the plurality of distinct terminal screens, wherein the remote computer system includes a Web application that includes a Web application server and an emulation services manager that is configured to: receive a client request for the existing menu-based legacy application, the request including a task identifier and input parameters; find a task corresponding to the client request; if the task is already executing, switch to providing output from a corresponding session from a session pool, otherwise allocate a new session from the session pool and start up the corresponding task; and interface via an emulation protocol with the existing menu-based legacy application.