Set 4 Space Indent in Emacs in Text Mode

Update: Since Emacs 24.4:

tab-stop-list is now implicitly extended to infinity. Its default value is changed to nil which means a tab stop every tab-width columns.

which means that there's no longer any need to be setting tab-stop-list in the way shown below, as you can keep it set to nil.

Original answer follows...

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It always pains me slightly seeing things like (setq tab-stop-list 4 8 12 ................) when the number-sequence function is sitting there waiting to be used.

(setq tab-stop-list (number-sequence 4 200 4))

or

(defun my-generate-tab-stops (&optional width max)
  "Return a sequence suitable for `tab-stop-list'."
  (let* ((max-column (or max 200))
         (tab-width (or width tab-width))
         (count (/ max-column tab-width)))
    (number-sequence tab-width (* tab-width count) tab-width)))

(setq tab-width 4)
(setq tab-stop-list (my-generate-tab-stops))

Short answer:

The key point is to tell emacs to insert whatever you want when indenting, this is done by changing the indent-line-function. It is easier to change it to insert a tab and then change tabs into 4 spaces than change it to insert 4 spaces. The following configuration will solve your problem:

(setq-default indent-tabs-mode nil)
(setq-default tab-width 4)
(setq indent-line-function 'insert-tab)

Explanation:

From Indentation Controlled by Major Mode @ emacs manual:

An important function of each major mode is to customize the key to indent properly for the language being edited.

[...]

The indent-line-function variable is the function to be used by (and various commands, like when calling indent-region) to indent the current line. The command indent-according-to-mode does no more than call this function.

[...]

The default value is indent-relative for many modes.

From indent-relative @ emacs manual:

Indent-relative Space out to under next indent point in previous nonblank line.

[...]

If the previous nonblank line has no indent points beyond the column point starts at, `tab-to-tab-stop' is done instead.

Just change the value of indent-line-function to the insert-tab function and configure tab insertion as 4 spaces.