programming - How to leave function unevaluated programmatically?

Is it possible to write a function in such manner, that it can be either evaluated or left unevaluated?

This is a general question. Sample below is just a sample. No need to find solution for this sample. Need a general solution.

Hypothetical sample:

F[x_] := If[x>=0,
    Sqrt[x],
    Fail[]
];

So that it gives

F[4]
(* ==> 2 *)

But

F[-4]
(* ==> F[-4] *)

I.e. hypothetical function Fail[] stays for "drop evaluation and leave expression unevaluated".

I know I can use Condition

F[x_] := Sqrt[x] /; x >= 0

But I wish to do that in imperative programming way. Is this possible?

UPDATE

I don't understand why this is not working:

F[x_] := Block[{fail},
  (If[x >= 0,
     fail = False;
     Sqrt[x],
     fail = True
     ]) /; ! fail]

UPDATE 1

Manual has a good example

f[x_] := Module[{a}, a = Prime[10^x];

  (FactorInteger[a + 1]) /; a < 10^6]

which is checking internal variables, not arguments. But why my version above not works then?

Answer

Your posted code does not work because some of the code you want to evaluate is given as an argument to Condition which holds its arguments, you could make it work by changing this

f[x_] := Block[{ans, success},
    ans = If[x >= 0, success = 1; Sqrt[x], success = 0];
    ans /; (success == 1)
    ]

Notice that the first line of code is process outside of Condition. It might be easier to see if you write it in expression form:

f[x_] := Block[{ans, success},
     ans = If[x >= 0, success = 1; Sqrt[x], success = 0];
     Condition[ans,success == 1]
     ]

Compare this with how your example would be in expression form:

 F[x_] := Block[{fail},
  Condition[If[x >= 0,fail = False;Sqrt[x],fail = True], ! fail]]

You should be able to tell the difference. In the working code the code that sets fail is evaluated before Condition is evaluated. In your code, Condition is evaluated first, which holds the evaluation of the code. This is because condition has Attributes HoldAll.