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I check my emails in Chrome using outlook.com, and occasionally I read through the emails marked as junk, both to see if they are junk, and to laugh at the amusing phishing attempts.
Could I be putting my computer at risk by opening up these emails? I never download any attachments or click any links.
TL;DR It's possible depending how your browser and email client handle files. Remote images are a potential risk with targeted attacks.
It depends on the security of your browser and other factors.
Risks of Remote Images There is the possibility your browser will view remote images. These images could be unique to that specific email and be used to let a spammer, phisher, etc read your message. They could get your IP Address and try to get more information this could be used as part of a targeted attack against you. Remember databases with IP Addresses get leaked all the time. If your attacker is smart enough and has the right resources they could steal your identity all with only an IP Address.
Browser Security / Web Client Security If you are using an outdated browser if your email provider does not properly strip malicious inline JavaScript (emails can use HTML therefore inline JavaScript) you could become infected if a security issue in your web browser exists.
Attachments Many attachments such as .docx files could be infected to install malware on your computer. You shouldn't open unknown attachments. You can scan first with VirusTotal even then please only open them at your own risk.
