To answer your question, yes, Illustrator is accurate for the size of a business card, or any print document.
The problem: 100% zoom is not the physical size of your document on-screen. Selecting that zoom level will not display the physical size, and even the "Actual size" zoom will not display the right size. The reason for this is that Illustrator knows nothing about your screen's pixel density, and this is what it would need to know to work out how much to zoom in so you could see a document at its physical size, and unfortunately there is no setting even to let Illustrator know this.
The Solution (a hack): Firstly, there's no need to print anything and waste paper/ink (let's save the planet and your money!). All you need to do is calibrate your zoom to match a physical sheet size, and if you want to, you can also store it as a new custom view, and set up a shortcut key so you can zoom in to that size any time you want.
How to do it:
Get a physical sheet of paper, either Letter size or A4
Create a new document the same size as the physical sheet you have.
Place the paper sheet against your screen, and type into the zoom field until the width on-screen matches the width of the sheet of paper. Take a note of this value.
At this point you can stop right here. Any time you want to see the phyiscal size of any document, just enter that zoom value into the zoom field. This value will be different for everybody, since the setting will depend on the pixel density of your monitor.
The following additional steps will allow you to set up a shortcut key to zoom to that size. This could be useful if you like to work zoomed in, but often want to zoom back to the physical size to check your document as you work on it.
Click View > New View - give it the name "Physical Size"
Open the keyboard shortucts using Edit > Keyboard Shortcuts
Change from Tools to Menu commands
Click on View
Scroll down until you see Custom View 1, and select the shortcut field
Type the new shortcut, something like Alt(Option)+Shift+Ctrl(Command)+Z will do the trick

Give your shortcut set a new name, to save it. Something like "My Shortcuts" will do.
Close the document, no need to save it.
Now to use the shortcut you just made with your business card document*:
Open your business card document
Type in the zoom you noted earlier in Step 3
Click View > New View. Name it "Physical size"
Save your document
Now the shortcut you created earlier will zoom to the physical size, and the custom view will be saved with your document. Also note that in the View menu, right at the bottom, the option "Physical size" can be selected if you forget the shortcut (which should also be displayed there).

*Note: You will have to repeat steps 12 to 15 for any new document you create. The shortcut is already created so you don't need to repeat the other steps above. They only need to be done once.