How to make Emacs show blank spaces

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.

As has been pointed out in a couple of other answers, the preferred method now is NOT to use smartindent, but instead use the following (in your .vimrc):

filetype plugin indent on
" show existing tab with 4 spaces width
set tabstop=4
" when indenting with '>', use 4 spaces width
set shiftwidth=4
" On pressing tab, insert 4 spaces
set expandtab

set smartindent
set tabstop=4
set shiftwidth=4
set expandtab

:help smartindent

:help auto-setting

:set all

:help C-indenting