I have a stack of rejection letters from decades ago, and a life-long annuity as a retired federal agent. "Law enforcement" generally includes those recognized as peace officers by the state and federal agents (or federal public safety officers) authorized to make arrests based on probable cause, serve search and arrest warrants, and carry firearms in the performance of their duties. There are investigative type agencies that employ those without these powers (FAA, CIA, DEA [prescription control, etc.], SEC, NSA, some Inspector General positions, etc.).
There are positions within the arena of criminal justice other than peace officer, like probation, corrections, juvenile justice positions, and possibly some positions more closely aligned with social work (like pretrial assessment). There are numerous books available that identify positions closely aligned with criminal justice. And, there are web sites, a quick check came up with: http://www.ask.com/bar?q=criminal+justice+careers&page=1&qsrc=178&ab=4&u=http://careers.cua.edu/handouts/criminaljusticecareers.htm.
Many major police and state and federal agencies employ criminal intelligence analysts. The number of these positions are generally limited, therefore they may be more difficult to get than a peace officer position. And, you will probably be competing against former military intelligence personnel. You may have to start at an administrative position and work your way into the intel position, or be a former law enforcement officer. All state and local agencies employ dispatchers and other administrative personnel.
A crime analyst may be deal with crime statistics (usually computerized) in such a manner as to provide decision-making advice on how to deploy personnel and implement programs designed to address crime problems. A criminal intelligence analyst looks at individual crimes or a specific type of crime in an effort to disclose linkages to people, places, or things. Some may do both.
Intelligence analysis consists mainly of review of documents, seeking connections and logical inferences in an effort to predict future events or establish prior associations. Whether or not it is interesting or boring varies from person to person, and may involve computer aided analysis. Intelligence analysis is generally divided into two classes, strategic and tactical. Strategic intelligence looks at the “big picture” with a longer-term focus, allowing prediction of future enforcement needs; while tactical intelligence concerns more immediate activities of specific offenders or organizations.
Some universities offer degrees in intelligence analysis. The American Public Univ System (apus.edu) is an accredited online university that offers 4yr and graduate degrees in intelligence and public safety (pertinent to physical security occupations).
There are also physical security positions in government and in the private sector that do not have physical fitness requirements. Any private company of any sized either contracts for or employs security of some sort. ASIS International (asisonline.org) is a private security organization that provides certifications in several areas of security.
Each state has a process for licensing private investigators, check the internet to identify the appropriate state agency (Dept of Public Safety in TX) for licensing requirements, and for a state PI association that sponsors training. A licensed agency may employ individuals as trainees.