Say I use Chrome on a new computer to log in to a website, and I select the option to have the browser remember the password. Then I download a straight-up smorgasbord of viruses of every kind. Can the viruses access the passwords that have been saved by my browser?
You should assume so no matter what*
When your system gets compromised you must assume that it has compromised any accounts stored on the system. You should go about changing the passwords for any accounts used on that computer in any place. You have no control over the virus, and you have no control how programs store your data so you don't know if your accounts are safe. As the general rule always assume a compromised computer means compromised accounts.
Password storage in browsers
For most browsers those passwords are stored encrypted, but since the computer is on and the infection has access to memory space it can find the key and decrypt them itself. These infections usually target the browsers specifically, but can also have a wider scope of impact.
The real attack the encrypted passwords protects against is the theft of the hard drive. However if the cipher text is in the memory then it can be found and taken.
Regarding Master Passwords and Password Managers
While convenient, these offer little protection from an infection due to the fact that the cipher text is still loaded into the memory space. These provide protection from sit down attacks, stolen hard drives, and user ineptitude.
They do not protect against an infection that has access to memory space
*: unless you know absolutely how and what and where the infections do their work.
There are methods of stealing browser-saved passwords so you need to assume that your browser is always vulnerable.
At a very minimum, you should set a browser master password and enable encryption to stored passwords. You can also look into using something like KeePass which allows you to store encrypted passwords where you choose (i.e. hard drives, portable drives, and network drives, etc).
Google Chrome, on it's own, is not designed to be very secure. LastPass is another half way decent option for securing browser passwords. LastPass gives you an added layer of protection as you need a master-password to start using it every time. And one of the most important aspects of using LastPass would be that it doesn't show the saved passwords on websites as it is unless and until you initiate LastPass, unlike password keeper.
This may lead to someone changing the form text box type to 'text' to view the contents in your saved password. This can be done using the inspect element feature of your browser.