I thought turning syntax on/off at appropriate places would make everything work, but I can't use ExplSyntaxOn/Off inside NewDocumentEnvironment so the following code sample doesn't work:

documentclass[varwidth]standalone
usepackageexpl3
usepackagexparse
begindocument
 NewDocumentEnvironmentmyPar
 
 parbox40mm
 % START PARBOX
 ExplSyntaxOn
 dim_new:NmyDim % DOESN'T WORK
 ExplSyntaxOff
 
 
 % END PARBOX
 
 beginmyPar
 Wikipedia is hosted by the Wikimedia Foundation, a non-profit organization that also hosts a range of other projects.
 endmyPar
enddocument

So I have to enclose the environment within syntax switches, as follows, but then anything passed to the environment loses its spaces:

documentclass[varwidth]standalone
usepackageexpl3
usepackagexparse
begindocument
ExplSyntaxOn % BROUGHT OUTSIDE
 NewDocumentEnvironmentmyPar
 
 parbox40mm
 % START PARBOX
 dim_new:NmyDim % WORKS (BUT NO SPACES REMAIN)
 
 
 % END PARBOX
 
ExplSyntaxOff % BROUGHT OUTSIDE
 % SPACES OF THE FOLLOWING CONTENT ARE STRIPPED
 beginmyPar
 Wikipedia is hosted by the Wikimedia Foundation, a non-profit organization that also hosts a range of other projects.
 endmyPar
enddocument

How do I retain spaces of the passed content without replacing them with ~ symbols?

By the way, following doesn't strip spaces (but lacks parbox as I need it):

documentclass[varwidth]standalone
usepackageexpl3
usepackagexparse
begindocument
ExplSyntaxOn % BROUGHT OUTSIDE
 NewDocumentEnvironmentmyPar
 
 dim_new:NmyDim % WORKS (NO SPACES STRIPPED BUT NO PARBOX)
 
ExplSyntaxOff % BROUGHT OUTSIDE
 % NO SPACES STRIPPED FROM THE FOLLOWING CONTENT (BUT NO PARBOX)
 beginmyPar
 Wikipedia is hosted by the Wikimedia Foundation, a non-profit organization that also hosts a range of other projects.
 endmyPar
enddocument

You are tricking yourself with improper indentation. Remember that TeX does not care about indentation; any closing brace will end the group started by the last opening brace. Let's look at the environment definition in your second code snippet with the correct indentation (i.e. indentation that reflects how TeX reads your code).

ExplSyntaxOn % BROUGHT OUTSIDE
 NewDocumentEnvironmentmyPar
 
 parbox40mm
 % START PARBOX
 dim_new:NmyDim % WORKS (BUT NO SPACES REMAIN)
 
 
 % END PARBOX
 
 ExplSyntaxOff % BROUGHT OUTSIDE

As you can see, your parbox ends right after the dim_new:N line, then there is an empty group and then the group of the environment start code ends. Importantly there is no environment end code before ExplSyntaxOff, which means that ExplSyntaxOff itself is taken as the environment end code.

Thus, expl3 syntax stays on, which causes all the spaces to be gobbled (among other havoc). This is not isolated to your environment. Add any text outside of beginmyPar .. endmypar and you will see the same problems. This is even true after the environment because the ExplSyntaxOff at the end of the environment is confined to the environment group and has no effect outside of it.

As Ulrike Fischer pointed out in her comment, using dim_new:NmyDim inside the environment is also problematic. You will get an error message telling you that myDim is already defined once you use the environment a second time. Define myDim once in the preamble and then only use it inside the environment.

You can avoid your grouping problem by using the b-type argument as demonstrated in Henri Menke's answer or just using a minipage instead of a parbox:

ExplSyntaxOn
 NewDocumentEnvironmentmyPar
 beginminipage40mm
 % Don't do dim_new:NmyDim here!
 
 endminipage
 
ExplSyntaxOff

1. Unbalanced token lists

Here's what you're trying to do is, in traditional terms,

newenvironmentmyPar
 %
 parbox40mm% start parbox
 
 %
 % end parbox
 

This is wrong code to begin with. Written in a different way to better see what happens

newenvironmentmyParparbox40mm

This is unfinished, because it misses the third mandatory argument to newenvironment.

It is not possible to have a brace unbalanced token list in the argument to a macro. The } that you mean to balance the before parbox is instead matching the after parbox. Chaos ensues.

2. Internal syntax markers

Whoever wants to delve into LaTeX programming should know about tokenization. When input is absorbed by TeX, it is transformed into tokens: symbolic tokens (vulgo, commands) and character tokens. The latter have their category code permanently attached. So if you say

newcommandfoomakeatletter @makeatother

you get @ to have the category code it has when newcommand is performed. So if you do

newcommandfoomakeatletterdimen@=2p@makeatother

upon execution of foo you'll receive puzzling error messages, unless the definition is surrounded by makeatletter and makeatother itself. But then the ones inside the definition do nothing at all. See What do makeatletter and makeatother do? for more information.

It is exactly the same with ExplSyntaxOn and ExplSyntaxOff.

What happens in your second attempt, when bringing them outside the code for myPar is that, due to the mistake described in chapter 1, ExplSyntaxOff is taken as the “end part” of the myPar environment, so it is not executed and ExplSyntaxOn remains in force when reading the text following beginmyPar.

3. Miscellaneous

Doing dim_new:N as part of the environment is wrong no matter what. Doing newlength inside an environment would be wrong as well, although it doesn't raise errors. Let's see with an example

newenvironmentfoonewlengthfoolen

will dutifully create and waste a new skip register at each call of beginfoo. This is because the name assignment is local, but the register assignment is global as LaTeX doesn't want to take the risk of referring to the same register in different ways.

With expl3 the things are a bit different: with

NewDocumentEnvironmentfoo
 dim_new:N myDim
 

(properly surrounded by ExplSyntaxOn and ExplSyntaxOff), also the name assignment is global. Thus upon calling beginfoo a second time LaTeX will raise an error.

4. How to make a parbox with an environment?

Use minipage, of course.

Use the new b-type argument which captures the body of an environment for use like a regular argument #1. The +b allows multiple paragraphs.

documentclass[varwidth]standalone
usepackagexparse

ExplSyntaxOn

NewDocumentEnvironment myPar +b 
 
 parbox 40mm 
 
 dim_new:N myDim
 #1
 
 

ExplSyntaxOff

begindocument
beginmyPar
 Wikipedia is hosted by the Wikimedia Foundation, a non-profit organization that also hosts a range of other projects.
endmyPar
enddocument