oh.. and I forgot one important file 😀 it is a cd image cd.iso which is mounted by dosbox prior to running the game

http://store.steampowered.com/app/7660/

Dosfreak, there was a Humble Bundle about a month or two ago that had drm-free X-Com games in it.
The X-Com Apocalypse I bought from that has a nearly identical layout except for having a capture sub-directory.

Hadn't tried it till now. And didn't have any problems with saving.

What sort of permissions does the SAVEGAME sub-directory in XCOMA have? You could try creating a file in there to see if you have write permission.
Is it mounted somewhere that you do not have write permission, or created when logged in as a different user?
This is a long shot, but does the mount c have some reference to being mounted as a CD?
Here is the autoexec part of the custom dosbox.conf that came with the game
@echo off
mount c . -freesize 1024
imgmount d cd.iso -t iso
c:
cd xcoma
xcomapoc.exe
exit

I always loved XCOM Apocolypse. Many years ago I tried running it on DOSBOX to no avail. I also tried loading a virtual machine with DOS 6 and Windows 95 but could not get it to load properly and run. I recently saw that Steam had a DOSBOX solution and it was cheap so I purchased it and was making good progress. The game worked exceptionally well. I am quite far into the game and I have built 3 bases. Now every time I go into base management and click on Base 2, the game crashes. I have contacted Steam who forwarded me to 2k.com for support. They tried Googling the answer but found nothing. They did not ask for logs or anything... They suggested to change the graphic settings (Which made the normal game just not work) and they suggested to switch off the sound emulation and play the game without sound, which just made the game quiet, not a fix. I have tried loading earlier save games, but eventually Base 2 gets 'large' and the game crashes again. The errors look like memory errors or buffer errors, where the code is writing to places it should not.

I then decided to download DosBox for myself and configure it. I do own the original game on CD. I have managed to do all this just fine. I copied my saved games from the Steam installation and immediately I get the same issues. I have increased memory and reduced it. (Below 4 MB and the game won't start) I found a mount command where it suggests you use a mount -freesize 1024 to show disk capacity of 1024 MB free or else it shows about 200MB which might not be enough for you save games... did not work. I decide to try play with the CPU settings which are set to Auto by default, and the 386 setting actually generates the same error as I am getting but during the startup process, before I can even get the start menu. So I am wondering if this is not a CPU emulation error, but why based on file size or complexity...?

Other than this the game work just perfectly... I just cannot continue my game further. Any help or advise will be appreciated. Even if I need to load a Linux kernel and run Wine or some other emulator... Would it be worth my while to try load earlier versions of DosBox? (if so, which one?)

XCOM has an extensive community; I would suggest looking at the various XCOM fan forums for an answer. Sometimes DOS4GW errors can be resolved by substituting DOS32a, but if that was a solution then someone out there would have posted about it before now.

Probably the best solution would be to run the game using Open XCOM. This is a relatively recent development.
http://openxcom.org/