Can I get hacked if I don't download anything?

There are all sorts of viruses, spyware, and even browser exploits that are 'drive by' infections you get just by visiting a page.

The way to prevent this 100% is to run the OS in a non-administrator mode (logged in as a non-priviledged user or using a read-only OS like a live Linux distro) AND/OR use a script-blocking add-on like NoScript. Scripts are what infect your computer, so if you block scripts from executing, you reduce the chance some script will install malware.

Web pages have analytics built into them, and unless you are using some sort of anonymizer or proxy like the TOR service, your IP address is very trackable. Some browsers also have a
private mode that blocks some of the analytics, but the IP address can still be identified, although it's trickier to do.

I manage a web site, and I load both Google Analytics and TraceMyIP analytics on each page...I can see lots of info about who visits my site..IP address, location, OS, browser...etc.

Also, there are people out there that just ping every IP address to look for computers attached directly to the web.
Usually running a router protects you from these threats.

One example is an old windows 95 virus that can only infect you if your C drive is shared. You don't ever need to open your browser. You will just randomly get this virus.
I picked it up again last year on a VM so it must still be circulating. It usually adds fake screensavers.

You asked: "someone I know online... Is it still just as easy for them to hack my computer?"

The answer is, No.

"Hacking your computer" would imply installing some malware or virus/tracking or surveillance software, or somehow setting up some sort of remote connection.

In terms of installing any sort of malware, the same protections apply....not clicking on unknown links or downloading unknown attachments without scanning them, keeping PC patched and using good AV software, etc. etc.

In terms of somebody gaining remote access, normally people have a firewall which does not allow inbound connections. So if, for example, you had remote desktop enabled on your PC, unless the port required for that (port 3389, typically) was open on your firewall/router an outsider cannot connect.

So to recap, for an individual to hack you, they would either have to get some sort of program or malware on your PC, or you would have to have some port or ports open on your firewall. If you are careful with respect to viruses and malware, the former won't happen, and if you have a standard home router/firewall, these are configured with all ports closed by default, so most likely the latter won't happen either.

To clarify one response given above....if you use an email CLIENT such as Outlook Express, Thunderbird, Apple Mail, then the originating IP address of the email will be where it came from (e.g your home IP.

However, if you use webmail (gmail, yahoo, etc), then the originating IP address will be that of the webmail service ( gmail, yahoo), not yours.