By executable, you probably mean the
.exe "executable" format that Windows uses to denote anything that contains a program. (Note that other extensions such as
.com may also be executable, and it are not only executables that can get you infected; a malicious PDF does the job just as well.)
Any OS allows the execution of files, or else you can't do things like starting Powerpoint. You are right though that Chrome OS can't (by default) execute Windows' .exe files, but it's just as easy to make "binaries" (a more general term for executable files) for Chrome OS.
Chrome OS is based on Linux, making it vulnerable for pretty much any attacks that work on Linux computers, though it seems some things are sandboxed more than most other Linux distributions would do (see rakslice's comment). Many things in Chrome OS are done in the browser which makes it more secure, but this still doesn't eliminate all exploits. Migrating to Chrome OS would at least temporarily solve your problem since any attackers need to rethink their strategy, but it won't take very long. Windows, when used correctly, is a reasonably safe OS to use. The fact that you have real security issues (as you put it) hints at an underlying problem such as incompetence of employees (opening untrusted email attachments), bad patching, or something else.
If the attacks are by chance, e.g. when a user visits a website that happens to be infected, moving to Chrome OS (or any other Linux system) will solve your problem. In small organisations this might be an option, though in general I wouldn't advise to change the OS just because some employees happened to visit an infected website. Better install good anti-virus software, monitor workstations, limit permissions as much as possible, and setup a plan to perform when infections do occur. Something like: the computer will be taken off the network, the harddrive will be inspected and scanned in a safe environment, when the malicious part is found other computers will be scanned to see whether they have the same, and any infected computers will get a complete reinstall (shouldn't be too much work with system images and adequate backups).
If the attacks are targeted at your company specifically, you're in bigger trouble. Get someone on board with real knowledge of security, at least temporarily to get things under control. Targeted hacking attacks by professionals are hard to defeat and usually beyond the control of most sysadmins.