I have a large web site (1,000 pages) created with traditional
static-html tools. I do NOT want to make the whole thing php based,
but I do want to start allowing some people to edit pages freely while
keeping an ability to see and approve changes individually. I do not
have an unlimited software budget, and the server is a standard
Linux-Apache-PHP-MySQL-Perl setup (PHP 4). Is there any software out
there that would let me do this without going crazy?

It may be a good idea to leave what you have alone, and create a wiki
site where you can just copy and paste all of the pages you want, have
them work there.
http://www.jot.com/request_setup/?utm_source=google&utm_medium=ppc&utm_term=Wiki%20server&utm_campaign=general_wiki_terms

Once there just click on the links on the top right hand for more info.

Good luck.
Elí

Though I appreciate the thought, and have considered it, that's not
really what I was looking for; I want to keep the same basic structure
and allow people to add to it under tightly controlled conditions.

There are wikis that allow you to define page styles, users, access
control and the such. (try looking for wakka or derivatives, such as
uniwakka)
The other easy option is to use a CMS (one for all: plone).

A wiki sounds good for new or small sites just starting up, but
converting a thousand-page site into a wiki would be a rather large
udnertaking - also, the dynamic serving of pages would require more
server strength, I suspect. In addition, most pages are not changed
very often; and I need to have different headers in different sections
for advertiser flexibility. If a wiki can handle that, I'm all for it.
Otherwise I'd need a different solution. I've been thinking about
those programs that use editable areas but haven't found any that have
an editing+approval process or CVS.

...though OpenCMS does look interesting since it can import entire
sites and export to static. Wakka which you recommended is no longer
being developed. Plone ... hmmm/

One more clarification: I need to keep all URLs the same as they are
now due to incoming links and Google ranking consideratons.

Hello davert3, in this case, you can install and configure a CVS
server to be able to make a pursuit of all the changes made to each
page and to return to previous versions of each page if you want.

You can find much aid by reading this book:

http://cvsbook.red-bean.com/

Additionally I recommend to you to use graphical CVS clients  to
facilitate the work of your users.  I recommend the following ones to
you:

http://www.jcvs.org/
http://www.wincvs.org/

Regards.

QBert

I will investigate those, thanks.

qbert2k-ga, I need you to submit that as an answer so you can get paid!

PS> Your answer is good but I think CVS is going to be way too complex
... will still pay you for this if you enter it as an answer.